Research Seminar in Mathematics - Lefschetz properties of monomial ideals

Speaker

Nasrin Altafi Razlighi.

Abstract

An artinian graded algebra is said to satisfy the strong Lefschetz property, (SLP) if multiplication by all powers of a general linear form has maximal rank in every degree. If this property holds for the first power the algebra is said to have the weak Lefschetz property (WLP). The study of the SLP has its origin in the Hard Lefschetz theorem. In fact the cohomology ring of a complex projective smooth variety has the SLP. There have been a lot of studies on algebras satisfying or failing the Lefschetz properties. In this talk, we will present some of the main results and  talk about bounds on the Hilbert function of the monomial artinian algebras failing the WLP. We will also study the WLP of artinian ideals in a polynomial ring generated by invariant forms under an action of the cyclic group and give a complete classification of such ideals satisfying the WLP in terms of the action.